A parabola with equation y^2=4ax for constant a is translated by the vector (2,3) to give the curve C. The curve C passes through the point (4,7), what is the value of a?

Invert the translation of (2,3) to get the parabola passing through the point (4,7)-(2,3)=(2,4). This is the same as saying that y=4 when x=2, substitute this into your equation y^2=4ax to get a=2.This will be seen easier with a picture of the parabola and the curve C.